= Representation ring of an odd-dimensional special orthogonal group
{title2=$R(SO(2m+1))\cong\mathbb Z[z_1^{\pm1},\ldots,z_m^{\pm1}]^{(\mathbb Z/2)^m\rtimes S_m}$}
For the block-rotation <maximal torus> in the <special orthogonal group> $SO(2m+1)$, the <Weyl group> acts by permutations and independent inversions of $z_i$. Its invariant <Laurent polynomials> are the symmetric polynomials in $x_i=z_i+z_i^{-1}$, hence the polynomial ring in their <elementary symmetric polynomials> $e_1,\ldots,e_m$. Restriction of the group <representation ring of a compact group> is injective because every group element is conjugate into the torus and <characters> distinguish virtual representations. For the defining complexified representation $V$, the exterior-power characters satisfy $\sum_k\chi_{\Lambda^kV}t^k=(1+t)\prod_i(1+x_it+t^2)$. For $k\leq m$, its coefficient is $e_k$ plus an integral linear combination of $e_j$ for $j<k$. Induction puts every $e_k$ in the restriction image, proving equality with the invariant ring and giving polynomial generators $[\Lambda^1V],\ldots,[\Lambda^mV]$.
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